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round a branch point. To this circuiting of a "branch point on the
single z-plane corresponds, on the n-sheeted Riemann surface
the passage from one sheet to another and the resultant inter-
change of the branches of the function.
There are many ways in which the variable may wander
about the n-sheeted Riemann surface, passing from one sheet to
another, To each of these corresponds a particular interchange
of the branches of the function, which may be symbolized by
writing, one after another, letters denoting the several branches
interchanged. In this way we get the symbols of certain
substitutions (as in chapter 15) on n letters; all of these substitu-
tions generate a group which, in some respects, pictures the
nature of the function considered.
Riemann surfaces are not easy to represent pictorially, and
those who use them content themselves with diagrammatical
representations of the connexion of the sheets, in much the
same way that an organic chemist writes a 4graphicaP formula
for a complicated carbon compound which recalls in a schematic
manner the chemical behaviour of the compound but which
does not, and is not meant to, depict the actual spatial arrange-
ment of the atoms in the compound. Riemann made wonderful
advances, particularly in the theory of Abelian functions, by
means of his surfaces and their topology - how shall the cuts be
made so as to render the n-sheeted surface equivalent to a
plane, being one question in this direction. But mathematicians
are like other mortals in their ability to visualize complicated
spatial relationships, namely, a high degree of spatial "intuition*
is excessively rare.
Early in November, 1851, Riemann submitted his doctoral
dissertation, Grundlagen fur eine allgemeine Theorie der Funk-
tionen einer verdnderlichcn complexen Grdsse (Foundations for a
general theory of functions of a complex variable), for Gauss*
consideration. This work by the young master of twenty-five
was one of the few modern contributions to mathematics that
roused the enthusiasm of Gauss, then an almost legendary
figure within four years of his death. When Riemann called on
Gauss, after the latter had read the dissertation., Gauss told Mm
that he himself had planned for years to write a treatise on the